In this subsection, we will adopt the definition of boundedness of a set-theoretic family of sheaves in [[1]}, Definition 4.10].
Since an \(S\) -flat coherent sheaf \(\mathcal {H}\)  on \(\mathcal {X}\)  is also a set-theoretic family of coherent sheaves on \(\mathcal {X}\) , which is obviously bounded. By Kleiman criterion for stacks in [[1]}, Theorem 4.12], there exists an integer \(m\)  such that
\(\mathcal {H}_{s}\)  is \(m\) -regular for every geometric point \(s\)  of \(S\)  (see [[1]}, Definition 4.1 and 4.2] for \(m\) -regularity of \(\mathcal {H}_{s}\)  on \(\mathcal {X}_{s}\) ). As in [[4]}, Definition 1.7.3], we define the regularity of an \(S\) -flat coherent sheaf \(\mathcal {H}\)  as
\(\mathrm {reg}_{\mathcal {E}}(\mathcal {H}):=\inf \lbrace m | \mathcal {H}_{s} \mbox{ is $m$-regular for every geometric point $s$ of $S$}\rbrace .\) 
